Now that you have investigated one type of polygon (rectangles), let's see if the relationship between the scale factor of a dilated polygon and the area holds true for other polygons and circles as well.

Recall that the perimeter of a circle is called the circumference.

Click on the image below to access an activity with similar triangles. Move any point on ΔPQR to dilate it. The scale is located in the upper left hand corner.  (Remember from the first section that the scale factor is the ratio of corresponding sides.)  Notice how the dilated area and the ratio of areas change.  Fill in the following chart.

Click here to see a completed table

Area of ΔLMN	Scale Factor	Area of ΔPQR	(Scale Factor)2	Area of ΔLMN × (Scale Factor)2
9.2	3	83.1*	9	82.8*
9.2	2	36.9	4	36.8
9.2	1.5	20.7	2.25	20.7
9.2	0.75	6.9	0.5625	5.175
9.2	0.5	2.3	0.25	2.3

*The answers are close; the applet rounds creating a slight difference.

What is the relationship between the area of ΔLMN, the scale factor, and the area of ΔPQR?

Check Your Answer

The area of ΔPQR is equal to the product of the area of ΔLMN and the square of the scale factor.

Click on the imiage below to see an activity that will find the area of a circle. By changing the radius (moving the orange dot), the circle dilates. Use the activity to complete the following table.  Start with a circle with a radius of 1.

Click here to see a completed table

Area of Original Circle	Scale Factor	Area of Dilated Circle	(Scale Factor)2	Area of Original Circle × (Scale Factor)2
3.1	3	28.3*	9	27.9*
3.1	2	12.6	4	12.4
3.1	2.6	21.2	6.76	21.0
3.1	1.8	10.2	3.24	10.044
3.1	1.5	7.1	2.25	7.0

*The answers are close; the applet rounds creating a slight difference.

What is the relationship between the area of the original circle, the scale factor, and the area of the dilated circle?

Check Your Answer

The area of the dilated circle is equal to the product of the area of the original circle and the square of the scale factor.

Practice